C 2006)
Journal of Abnormal Child Psychology, Vol. 34, No. 3, June 2006, pp. 379–391 (
DOI: 10.1007/s10802-006-9027-x
The Short Mood and Feelings Questionnaire (SMFQ):
A Unidimensional Item Response Theory and Categorical
Data Factor Analysis of Self-Report Ratings from
a Community Sample of 7-through 11-Year-Old Children
Carla Sharp,1,4 Ian M. Goodyer,2 and Tim J. Croudace3
Received February 24, 2004; revision received May 10, 2005; accepted September 13, 2005
Published online: 29 April 2006
Item response theory (IRT) and categorical data factor analysis (CDFA) are complementary methods
for the analysis of the psychometric properties of psychiatric measures that purport to measure
latent constructs. These methods have been applied to relatively few child and adolescent measures.
We provide the first combined IRT and CDFA analysis of a clinical measure (the Short Mood and
Feelings Questionnaire—SMFQ) in a community sample of 7-through 11-year-old children. Both
latent variable models supported the internal construct validity of a single underlying continuum
of severity of depressive symptoms. SMFQ items discriminated well at the more severe end of the
depressive latent trait. Item performance was not affected by age, although age correlated significantly
with latent SMFQ scores suggesting that symptom severity increased within the age period of 7–11.
These results extend existing psychometric studies of the SMFQ and confirm its scaling properties
as a potential dimensional measure of symptom severity of childhood depression in community
samples.
KEY WORDS: Screening; childhood depression; SMFQ; item response theory; categorical data factor analysis.
Over the last 40 years, the methods used to evaluate
the psychometric basis of ability tests, health care surveys, and multi-item screening instruments has changed
dramatically. Whilst the methodology of classical test
theory (CTT) has served test development well, item response/latent trait theory (IRT) approaches have become
more mainstream as the technical basis for measurement
theory, test construction and scale evaluation (Embretson
& Reise, 2000). Although moves towards adoption of
more appropriate, non-linear and categorical data factor
analysis (CDFA) models have been most apparent in ed1
2
3
4
ucational settings, in the last two decades such methods
have begun to be applied in clinical testing of adults. This
has been evidenced by psychometric studies published
in, for example, Psychological Assessment and Psychological Methods (e.g. Angold, Erkanli, Silberg, Eaves, &
Costello, 2002; Cooke & Michie, 1997; Lambert et al.,
2003; Patton, Carlin, Shao, Hibbert, & Bowes, 1997;
Santor, Ramsay, & Zuroff, 1994). Currently there are
very few reports that have applied such methodologies
in samples of young children (Cheong & Raudenbush,
2000).
One reason for the under-exploitation of such
methodologies may be because researchers have not been
introduced to the potential and practicalities of these
methods (Rouse, Finger, & Butcher, 1999) and are therefore unaware of the advantages they offer over conventional (CTT) methods (van der Linden & Hambleton,
1997). Although CTT is often included in the curriculum of both clinical and applied psychologists, IRT is
rarely taught, and has had less coverage in mainstream
Menninger Department of Psychiatry and Behavioral Sciences, Baylor
College of Medicine, Houston, Texas.
Developmental Psychiatry Section, University of Cambridge,
Cambridge, UK.
Department of Psychiatry, University of Cambridge, Cambridge, UK.
Address all correspondence to Carla Sharp, Menninger Department of Psychiatry and Behavioral Sciences, Baylor College
of Medicine, 1 Baylor Plaza, Houston, Texas 77030; e-mail:
csharp@hnl.bcm.tmc.edu.
379
C 2006 Springer Science+Business Media, Inc.
0091-0627/06/0600-0379/0 380
psychology journals (Embretson & Reise, 2000). We provide an application of IRT and categorical data factor
analysis (CDFA) methods to a commonly used self-report
measure of depressive symptoms in children, the Short
Mood and Feelings Questionnaire (SMFQ; Angold et al.,
1995). As such, the aim of this paper was to scrutinize
the internal construct validity of the SMFQ. To this end,
we used latent variable models implemented with both an
IRT and appropriate factor analysis framework (CDFA).
Within a latent variable framework, internal construct
validity refers to an understanding of the psychometric
performance of items in relation to an underlying (latent) construct of interest. Here the latent variable was
a continuum of severity derived from self-reported depressive symptoms in 7–11-year-old children. This latent
variable was psychometrically derived, and was not validated through an external criterion measure of depression; hence our results pertain only to internal construct
validity.
The Mood and Feelings Questionnaire (MFQ), long
form, was developed as a screening tool for detecting
clinically meaningful signs and symptoms of depressive
disorders in children and adolescents (6–17 years of age)
by self-report (Angold et al., 1995; Costello & Angold,
1988). MFQ items were designed to cover DSM diagnostic criteria for major depressive disorder (APA, 1994).
Over the past decade, the long form consisting of 33 items
has been used extensively in both epidemiological and
clinical research (Costello et al., 1996a, 1996b; Goodyer,
Herbert, Tamplin, & Altham, 2000; Kent, Vostanis, &
Feehan, 1997; Park, Goodyer, & Teasdale, 2002; Wood,
Kroll, Moore, & Harrington, 1995). Criterion-related validity (ability to predict clinical diagnosis) has been established for the long form (Wood et al., 1995).
A short form consisting of 13 items (SMFQ) was
subsequently derived for which criterion validity has also
been shown (Angold et al., 1995; Kent et al., 1997; Thapar
& McGuffin, 1998). Table I presents details of studies that
have used the self-report version of both the long and short
form of the MFQ and summarizes the validity issues addressed. Although these studies are universally supportive
of the internal construct validity of the SMFQ, most samples were from clinic or specially selected populations.
Currently, there is no published study on the internal construct validity of the short version in a community sample of primary school-aged children. Establishing internal
construct validity in 7–11-year-old children is essential if
the SMFQ is to be recommended as a self-report screening
measure of severity of depression in community samples
of children.
Most existing studies, as summarized in Table I, have
applied statistical procedures based on CTT, linear factor
Sharp, Goodyer, and Croudace
analysis or principal components. Such methods are not
optimal for the discrete/categorical nature of the MFQ
responses (Angold et al., 2002) for several reasons. Traditional methods, such as principal components analysis,
assume that item responses are on a continuous metric, yet, psychopathology ratings are recorded using discrete categories in most community studies to date. Usually, the commonest (modal) response on SMFQ items
is zero, representing absence of symptoms. Even when
ratings are collected on graded scales, response distributions are usually heavily skewed. Failure to treat symptom ratings as categorical data in factor analysis models
has two consequences: (1) in multi-dimensional analysis the true factor structure may be severely distorted,
and (2) in unidimensional models factor loadings may be
biased and resultant (weighted) scores estimated incorrectly. When applying linear models to binary (0,1) data,
predictions are made that are not within the plausible
range (<0 or >1) (McDonald, 1999). This is obviously
highly undesirable when relatively rare symptoms are being modelled. The limitations of linear models in these
situations are well known to statisticians and methodologists, but are frequently ignored by applied researchers.
Only in limited circumstances will linear methods approximate a more appropriate model (Shrout & Parides,
1992).
Although CTT and IRT/CDFA both assume that variation in the observed responses to items of a test can be
explained by one or more continuous unobserved latent
traits, the way IRT/CDFA models the relation between
observed item-responses and the latent trait is different.
Instead of summarizing the psychometric properties of
a scale with omnibus statistics (such as item-total correlations or Cronbach alpha), thereby averaging across
levels of individual variation (Santor et al., 1994), IRT approaches model how the probability of responding to an
item—here this is equivalent to endorsing a symptom—
varies as a function of the location along a latent continuum or dimension of variation (Santor et al., 1994).
IRT methods do not use summary statistics that apply to
groups of individuals, such as correlations, but can define
a model for the individual response patterns that comprise
the raw data. Because item response patterns can be modelled directly within an IRT framework, no information in
the data is lost.
A mathematical equation—a probability model, similar to that used in logistic or probit regression—is used to
describe the non-linear relation between an item-response
and the value (severity) on the latent trait. This relation
can be represented graphically by a plot that is known
as an item characteristic curve (ICC) or item response
function (IRF). The ICC offers a graphical profile of item
An IRT and CDFA Analysis of the SMFQ
381
Table I. Summary of Validity Characteristics of Studies that Used the Self-report Versions of the MFQ Long and Short Forms
Author and date
Samples
Age
N Sex
Type of validity
assessed
Analytic strategy
10–19 43 boys
61 girls
ROC analyses
Criterion validity for
long form, using the
K-SADS as criterion
6–17
33 boys
15 girls
Internal consistency,
content and
criterion-related
validity for SMFQ,
using the DISC as
criterion
6–11
54 boys
71 girls
Messer et al. (1995) 1502 high risk
community children
6–13
Boys only
Internal consistency
Kent et al. (1997)
114 consecutive
attendees at four
clinics
7–17
56 boys
57 girls
Thapar and
McGuffin (1998)
411 twins
11–16 99 boy pairs
123 girl pairs
94 mixed pairs
Criterion validity for
long and SMFQ,
using the K-SADS
as criterion
Criterion validity for
SMFQ, using the
CAPA as criterion
Wood et al. (1995)
104 consecutive
referrals to
outpatient
psychiatric clinic
Angold et al. (1995) 48 consecutive
referrals to
outpatient
psychiatric clinic
125 referrals to a
pediatric clinic
Principle component
analysis
Cronbach’s alpha
Results
Sensitivity 0.78
Specificity 0.78
Unidimensional factor
structure (acceptable
model fit statistics)
Cronbach’s alpha 0.85
High ORs for
predicting
psychiatric
diagnosis
Sensitivity 0.60
Specificity 0.85
CFA using LISREL VII Unidimensional
structure of SMFQ
confirmed: GFI and
AGFI indices high
(>0.97); RMSR low
(>0.08); χ 2 /df
indices all <3
Correlational analyses Sensitivity 0.59
ROC analyses
Specificity 0.89
Maximum likelihood
logistic regression
ROC analyses
Correlational analyses
ROC analyses
Sensitivity 0.75
Specificity 0.74
Note. Criterion validity here refers to the SMFQ’s ability to detect major depressive episode in study samples with an acceptable degree of sensitivity and
specificity. Internal consistency refers to the underlying factor structure of the SMFQ, notably whether it can be demonstrated to be a unidimensional
scale. The current study did not aim to examine criterion validity but only internal consistency.
effectiveness, which is more informative than traditional
measures of item performance (Santor et al., 1994). For
a detailed discussion of the distinction between CTT and
IRT approaches and the benefits of using the latter, see
Embretson and Reise (2000).
More detailed information on the effective measurement range of individual items and scales is especially
important for psychopathology measurement in developmental epidemiology studies. Epidemiological evidence
has suggested that symptom profiles may differ with age
for reasons that are not fully understood. Weiss and Garber (2003) outlined several ways in which developmental
differences may impact on the phenomenology of depression over the course of childhood and adolescence. From
a psychometric perspective, it may be, for instance, that
certain items are not appropriate for certain age groups,
or that the symptoms (defined by the wording of questionnaire items) are not a feature of a particular disorder
in that age group. As the current study demonstrated (see
section on differential item function—DIF), IRT provides
the opportunity to distinguish between bias at the level
of the item (i.e. the item does not accurately probe for
the symptom for a particular age group) and bias at the
level of the latent trait (i.e. the disorder does not express
itself through a particular symptom in a particular age
group).
To our knowledge, only one study to date has
adopted an appropriate categorical factor analysis
382
approach to SMFQ data (Angold et al., 2002). Angold
et al. (2002) reported CDFA solutions estimated using weighted least squares estimation methods in Mplus
software. Their study confirmed the SMFQ’s unidimensional structure (single factor) in two samples: First,
the Great Smoky Mountains Study comprised 9-, 11and 13-year-olds. This sample of n = 1441 represented
25% of the highest scorers on the Child Behaviour
Checklist out of a community-based sample of n = 4500,
i.e. a “high-risk” sample. Similar results were found
in a second sample from a family study of n = 1412
twins.
Confirming the unidimensional factor structure of the
SMFQ with non-linear factor analytic techniques (Angold
et al., 2002) was an important first step in examining the
psychometric properties of the SMFQ from a latent trait
modelling perspective. However, no information was presented on the effective measurement range of the SMFQ.
This is more easily summarised using graphical representations of the model from an IRT perspective, i.e. the test
characteristic curve (TCC). We repeated and extended the
methods used by Angold et al. and offer more graphic representations of the latent variable modelling. In addition,
we provide the first IRT/CDFA modelling of SMFQ data
in a community sample of younger children (including 7and 8-year-olds).
Given concerns that there might be item bias with
respect to age (with different thresholds for response for
younger and older children), we further extended Angold’s approach by testing for item bias (DIF) using a
multiple indicator multiple cause (MIMIC) modelling approach (Gallo, Anthony, & Muthén, 1994). Item bias is
present when individuals with the same score on the psychometrically derived latent trait are more or less likely
to endorse an item. MIMIC modelling investigates item
bias through an extension of the CDFA factor analysis
model to include covariates, both of the latent trait, and of
the items. In testing for age invariance of item thresholds,
the estimates of interest are the direct effects (regressions
of) item responses on age as a covariate after adjustment
for the effect of age on the latent trait score. Given that
prevalence rates of depression increase with age in adolescence (Goodyer & Sharp, 2005; Hankin et al., 1998),
we expected this might also be the case for this age group.
We therefore anticipated finding an age difference in latent
trait scores, indicating more psychopathology in the older
children (10- and 11-year-olds), but no evidence of item
bias for any of the SMFQ items. We were able to test for
the effect of age, because unlike previously reported studies, our sample comprised a cohort of younger children
(7–11-year-olds) recruited and assessed in an elementary
school setting.
Sharp, Goodyer, and Croudace
METHOD
Participants
Parents of 2950, 7- to 11-year-old children (primary
school years 3–6) of 16 primary schools from a mixed
catchment of rural and urban areas in Cambridgeshire,
England were asked to participate. Response rates for
individual schools ranged from 14 to 40% resulting in
20% of the children taking part in the study (n = 659; 319
boys and 340 girls).
There are four possible reasons to explain the low
response rate. First, the ethics approval requirements prohibit researchers from gaining access to names and addresses of parents in the community. Invitation letters to
participate in the study were therefore handed out to children at school to take home to their parents. It is possible
that many invitation letters did not make it home in the first
place. Second, ethics in the UK require positive consent.
The effort of actually completing and returning a consent form to indicate positive consent may be too much
to ask of some parents in the community. Third, limited
resources precluded payment to children for their participation. Instead, children were given a sticker and were
entered into a school raffle drawing for their participation.
Fourth, it is possible that parents feel more protective of
children in the below-11 age range compared with adolescents, where the response rate for community studies
using a school-based ascertainment procedure in the UK
is typically 50% (Goodyer et al., 2000).
All children had an estimated IQ above 80. The mean
estimated IQ for the sample was 104 (SD = 14) and the
mean age was 9 years, 5 months (SD = 12 months). The
ethnic distribution in the sample was in line with regional
statistics (Office of National Statistics, 2001) for eastern
England (97% white, 2% of middle-eastern origin, 0.5%
black and 0.5% Asian). Two procedures were employed
to determine participation bias. First, permission was obtained for teachers of one of the schools to complete a
screening measure of common emotional and behaviour
problems, the Strengths and Difficulties Questionnaire
(Goodman, 1997, 2001; Goodman, Ford, Simmons, Gatward, & Meltzer, 2000) on all the children in the school.
Children who completed the SMFQ were compared with
those who did not for their ratings on the SDQ. Independent sample t-tests revealed no evidence of any differences
between the participants (n = 61) and non-participants
(n = 232) when the five sub-scales of the SDQ (hyperactivity, emotional symptoms, conduct problems, peer problems, prosocial behaviour) were compared. Comparison
of sociodemographic characteristics also revealed no difference between participants and non-participants.
An IRT and CDFA Analysis of the SMFQ
Measures and Procedure
Short Mood and Feelings Questionnaire
(Angold et al., 1995)
Our primary measure was the self-report SMFQ
which comprises 13 items with a common response format: 0, never; 1, sometimes; 2, always. The SMFQ was
administered individually at the same time as all the other
measures. Due to concerns raised regarding the reading
and understanding ability of younger children (Messer
et al., 1995; Thapar & McGuffin, 1998), teachers were
consulted as to the level of understanding for the 7-yearolds (youngest cohort), and it was decided that questions
would be read aloud to this group (8%). As in previous studies that have used the SMFQ with younger children (Angold et al., 1995), the answers recorded were the
participants’ self-reports and not the examiners’ opinions
about them. Children in higher grades were invited to ask
for help, if needed. However, none of the children in the
high grades did so.
IQ
A shortened version (Vocabulary and Block Design
subtests only) of the Wechsler Intelligence Scale for Children III (Wechsler, 1992), was used to estimate overall IQ
in the sample. Sattler’s (1988) guidelines were used for
administration and scoring.
Psychopathology
To assess response bias, teachers completed the
Strengths and Difficulties Questionnaire (SDQ; Goodman, 1997, 2001; Goodman et al., 2000). The SDQ was
specifically designed to screen for psychiatric disorders in
community samples and was shown to identify individuals
with psychiatric diagnosis with a specificity (the proportion of people without disease who have a negative test
result) of 94.6% (95% CI 94.1–95.1%) and a sensitivity of
63.3% (59.7–66.9%) (Goodman et al., 2000). Sensitivity
(the proportion of people with disease who have a positive
test result) for the SDQ has been demonstrated to be especially good for (70–90%) for identification of conduct,
oppositional disorders and hyperactivity disorders.
Data Analytic Strategy
Combining CDFA with IRT
Our primary data analytic strategy was to apply two
types of latent variable measurement models to the data:
383
Categorical data factor analysis and item response (latent trait) theory (IRT). In the literature, applications of
CDFA methodology often report only numerical results,
whereas applications of related IRT models summarise
their findings graphically. We wished to exploit both representations and therefore applied software for estimating
both types of models.
Background to Statistical Models
Introductory, technical and statistical accounts of
the family of the models that comprise the IRT and
CDFA approach are presented elsewhere (Baker, 2001;
Duncan-Jones, Grayson, & Moran, 1986; Lord, 1980;
Lord & Novick, 1968; van der Linden & Hambleton,
1997). Bartholomew and Knott (1999) and McDonald
(1999) provide excellent accounts of relations between
these models and more traditional models.
Briefly, CDFA is an extension of the linear factor
analysis model intended for dimension reduction and scaling of multiple binary or ordinal items. We consider only
the probit-probit factor model of Muthén (1984, 1989).
The model provides regression estimates of factor scores
which are location values for latent trait scores along the
single dimension of depression severity.
IRT analysis allows for detailed examination of the
properties of individual items by determining item characteristic curves. ICCs characterize the interaction of a
person with an item and plot the probability of a response
(endorsing a symptom) given the level of the underlying
characteristic measured by the “test” (Cooke & Michie,
1997)—in this case the severity of self-reported depressive
symptoms. ICCs are defined in terms of two parameters
that govern the shape and position of the S-shaped curves.
In educational testing settings where IRT models were
developed, the first parameter, a, is referred to as the item
“discrimination” and governs the steepness of the slope
of the ICC at the inflexion point of the S-shaped curve.
A lower a-value is associated with a more gradual slope.
Items with low slope estimates provide information over
a wider range of latent trait values. Items with higher
a-values have steeper curves and therefore discriminate
more finely, but over a smaller range of latent trait values.
The second parameter, b, relates to the prevalence of the
item (here the proportion of the sample that endorse each
SMFQ symptom). In educational settings, it is referred
to as the “difficulty” parameter since there it is related to
the difficulty of the test item. The b-parameter is related
to the point on the trait dimension at which a respondent
has a 50% probability of endorsing the item. In clinical
and epidemiological studies, this parameter is referred to
384
Sharp, Goodyer, and Croudace
as “commonality,” since it relates to the prevalence of
the symptoms. If the b-value is low, it indicates that the
item/symptom is frequently endorsed even among lowtrait individuals. In contrast, high values indicate that the
item/symptom is likely to be endorsed only among hightrait individuals (e.g. severely depressed).
of age on the location of the threshold term in the item
response model.
Graphical Representation of Scale
Performance from IRT
All participants who completed the SMFQ answered
all questions because questionnaires were individually administered and checked before they were returned. There
were 10 children who did not want to complete the questionnaire. Children were not required to give a reason
for non-participation. Full questionnaire data were therefore available for 649 children. There were no missing
data.
The SMFQ is a 13-item measure to which each response can be 2, 1 or 0. We recoded all responses to
binary format (0/1) because the low frequency (<5%) of
2 scores meant that very little information would be lost
in grouping these with 1 responses (options were recoded
in the following way: 0 = 0 and 1, 2 = 1). Another advantage of recoding to binary responses (collapsing the two
highest response categories) is that it provides a generally
more parsimonious model, and simplifies the reporting
of our IRT analysis of a candidate childhood depression
screening tool.
A 2 × 2 comparison (7-year-olds versus older ×
score of 0 versus 1 on the SMFQ) on each item showed
no significant differences between 7-year-olds (to whom
questions were read out) and the rest of the children. Children 8 years and older did not display difficulty completing the questionnaires by themselves. In addition, age did
not correlate with SMFQ total scores, although a significant relationship was found for SMFQ total scores and IQ
(r = −0.17; p < 0.01).
Table II shows the prevalence of binary recoded
item responses to each item: Five items were endorsed
by 15% or more of the sample (item numbers: 4, restless; 7, poor concentration; 3, tired; 10, felt lonely; 12,
never be as good); five items were endorsed by fewer
than 10% (item numbers: 2, not enjoy anything; 6, cried
a lot; 9, bad person; 11, unloved; 13, did everything
wrong).
The test information function (TIF) is a particularly
useful output of an IRT analysis. The TIF profiles variations in the precision of measurement of the latent trait
scores over the full range of estimated values (latent trait
scores). The TIF provides a compelling graphical assessment of the effective measurement range of the SMFQ
instrument. Reports of CDFA analyses do not usually
provide any information on the range over which reliable (accurate) scores are estimated for individuals, only
reliability at an aggregate level (for a sample group).
Software for Model Estimation
For the CDFA, we implemented Muthen’s robust weighted least squares estimation approach using Mplus software (see Flora & Curran, 2004). For
the IRT analyses, we implemented marginal maximum likelihood (MML) estimation of the two parameter logit-probit IRT model (Albanese & Knott,
1990; Bartholomew, 1987; Bartholomew & Knott, 1999;
Bartholomew, Steele, Moustaki, & Gabraith, 2002) in
TWOMISS. TWOMISS estimates model parameters for
the logit-probit model by a maximum likelihood procedure using a modified expectation-maximisation (EM)
algorithm.
MIMIC Modelling to Test for Differences in Item CDFA
Intercepts (thresholds) by Age
In order to explore the impact of age on item responses, we used the MIMIC modelling approach of Gallo
et al. (1994). This extends the categorical data factor
model to include a covariate. MIMIC modelling enabled
us to test the hypotheses of a linear association of latent
trait scores with age (as a continuous measure), but no impact of age on the SMFQ item intercepts (thresholds). The
presence of the latter would indicate differential item function with respect to age. We tested DIF for each SMFQ
item (one at a time) by freely estimating the correlation of
the latent factor with age, and also estimating the impact
RESULTS
Descriptive Statistics
Confirmatory Factor Analysis in Mplus Using
the Probit-Probit Model (Muthén, 1984, 1989)
First we summarize the results in terms of the lambda
and tau parameters from the confirmatory categorical
data factor analysis model. Then, we discuss indications of model fit [chi-square, root mean square error of
An IRT and CDFA Analysis of the SMFQ
385
Table II. Item Endorsement in Terms of the Proportion of the Sample that Endorsed Each Individual Item, Tetrachoric
Correlations Between Items, and Proportion of the Sample that Endorsed Pairs of Items
Item
1
2
3
4
5
6
7
8
9
10
11
12
13
%
1
0.107
0.094
0.176
0.275
0.114
0.090
0.178
0.114
0.060
0.164
0.088
0.151
0.077
0.331
0.412
0.233
0.474
0.439
0.490
0.474
0.589
0.523
0.525
0.474
0.495
2
3
0.023 0.043
0.036
0.356
0.060 0.181
0.448 0.335
0.362 0.403
0.510 0.401
0.473 0.357
0.209 0.258
0.361 0.284
0.261 0.435
0.440 0.037
0.380 0.497
4
5
6
7
0.045 0.036 0.028 0.049
0.029 0.031 0.022 0.046
0.065 0.040 0.037 0.063
0.054 0.036 0.091
0.315
0.031 0.049
0.188 0.468
0.031
0.430 0.456 0.299
0.373 0.686 0.468 0.531
0.286 0.507 0.447 0.565
0.175 0.599 0.478 0.455
0.341 0.594 0.386 0.478
0.239 0.662 0.388 0.526
0.370 0.551 0.554 0.628
8
9
10
11
12
0.036
0.032
0.042
0.059
0.054
0.031
0.056
0.028
0.011
0.020
0.029
0.025
0.019
0.036
0.028
0.049
0.034
0.049
0.060
0.059
0.040
0.065
0.051
0.031
0.032
0.017
0.039
0.045
0.039
0.022
0.042
0.032
0.023
0.048
0.043
0.037
0.056
0.062
0.062
0.032
0.068
0.054
0.028
0.057
0.043
0.567
0.509
0.500
0.578
0.600
0.509
0.557 0.593
0.477 0.449 0.559
0.493 0.397 0.470 0.618
13
0.028
0.020
0.039
0.042
0.032
0.028
0.048
0.036
0.019
0.031
0.023
0.043
Note. Column 1 (item) refers to the 13 SMFQ items. Column 2 (%) refers to proportion of the sample that endorsed each
item (responded 1or 2 rather than 0). The lower left triangle denotes the tetrachoric correlations between items. The upper
right triangle denotes the proportion of the sample that endorsed both items. Item labels, 1: Miserable/unhappy; 2: Not
enjoy anything; 3: Tired; 4: Restless; 5: No good anymore; 6: Cried a lot; 7: Poor concentration; 8: Hated self; 9: Bad
person; 10: Felt lonely; 11: Unloved; 12: Never as good; 13: Everything wrong.
approximation (RMSEA) and standardised root mean
square residuals (SRMR)].
Inspection of the factor loadings (column 3—lamda
values) for the full sample shows high values for all items.
Items 2, 3, 4 and 6 had the lowest factor loadings. The variance in item responses not explained by the single latent
factor is quantified by the residual variances in column 4.
Clearly these are quite substantial in magnitude for items
2, 3, 4 and 6. The standard errors (not shown) for these
items were also larger than for other items. Overall the
magnitude of most of the loadings is consistent with the
hypothesis underpinning our use of a single latent variable, that is, that the SMFQ items are relatively sensitive
(discriminating) indicators of an underlying continuum of
depression.
Tests of Model Fit
All indices of model fit (chi-square, RMSEA and
SRMR) supported the adequacy of a single latent variable model. The comparative fit index (CFI) (0.992) and
Tucker Lewis Index (TLI) (0.994) were both high and
close to 1; the root mean square error of approximation
was low (<0.06) (RMSEA = 0.018). The standardized
root mean square residual (0.063); the weighted root mean
square residual were also low (0.833). There was little scope for model fit improvement by increasing the
number of factors, and little justification in terms of our
study aims (full results available on request from first
author).
Full Information Item Factor Analysis Using the
Logit-Probit Model (Bartholomew & Knott, 1999)
Pattern Frequencies
We first considered pattern frequencies for all response patterns for the binary recoded data. Under the
full information approach, all information from the multiway cross-tabulation of all item responses is used. Thus,
there are 132 possible response patterns for which frequencies may be reported. In most samples with more
than six items, many possible patterns will not be observed
(since sample size is usually much less than the number
of itemsnumber of response categories ). Only with shorter instruments is it possible to display all pattern frequencies (i.e.
for scales with 5 or 6 binary items). For a 13-item questionnaire and a sample of n = 659 like the current study, 10
pages of response patterns were returned. Space does not
permit a full report of all response frequencies. Thus, only
the most common patterns will be commented upon here.
A full listing of response frequencies is available from
the first author (CS). The three most common responses
patterns are reported here.
Unsurprisingly, given the nature of the sample, the
most common pattern was the absence of all morbidity [0000000000000] with a frequency of n = 267 (41%).
This represents the modal pattern of children in the sample
who had none of the 13 indicators of depression.
The second most common pattern was
[0100000000000] with a frequency of 63 (10%)
386
Sharp, Goodyer, and Croudace
indicating endorsement of only one item, item 2 (did
not enjoy anything). The only other two-digit pattern
frequency was 23 (3%) for the pattern [0001110000000]
indicating endorsement of items 4 (restless), 5 (no good
anymore) and 6 (cried a lot). This is the modal response
for those children in the sample who had any symptoms
(at least one item endorsed).
median value on the latent trait. Here, these values range
from lower to upper limits. So far we have focussed
on the numerical results of the IRT and CDFA models. Next we consider the graphical representations that
further facilitate the interpretation of these parameter
estimates.
Parameter Estimates
Graphic Representation of Test
and Item Performance
Table III reports parameter estimates from
TWOMISS for the maximum likelihood factor analysis
under the Bartholomew and Knott (logit-probit) model.
Here, the IRT factor analysis model is parameterised differently, using alpha 0 for item intercepts and alpha 1 for
item slopes, but gives rise to very similar S-shapes for the
item characteristic curves (see later).
The final column of Table III reports a transformation of the alpha 0 parameter showing the probability of endorsing an item for an individual at the
Table III. Corresponding Categorical Data Factor Analysis Results
from Bartholomew and Knott’s (1999) Logit-Probit Item Response
Function Model
Item
1. Miserable
2. Not enjoy
3. Tired
4. Restless
5. Felt no good
6. Cried a lot
7. Poor
concentration
8. Hated self
9. Bad person
10. Felt lonely
11. Unloved
12. Never be as
good
13. Everything
wrong
Commonality Discrimination
α i0 (SD)
α i1 (SD)
Symptom
endorsement
probability for
median
individual, π i0
3.12 (0.29)
2.82 (0.231)
1.88 (0.152)
1.08 (0.105)
3.68 (0.432)
3.09 (0.274)
2.42 (0.246)
1.80 (0.28)
1.25 (0.23)
1.10 (0.17)
0.75 (0.13)
2.49 (0.40)
1.49 (0.26)
1.93 (0.27)
0.04
0.05
0.13
0.25
0.02
0.04
0.08
3.64 (0.441)
4.30 (0.541)
2.39 (0.223)
3.80 (0.476)
2.79 (0.275)
2.44 (0.41)
2.15 (0.43)
1.71 (0.24)
2.19 (0.42)
2.05 (0.28)
0.02
0.01
0.08
0.02
0.05
4.28 (0.545)
2.44 (0.46)
0.01
Note. Under the logit-probit model, the alphai0 (α i0 ) parameter in column 2 relates to the location on the x-axis of the inflexion point of
the item response function curve. Alphai1 (α i1 ) relates to the steepness of the slope of each ICC. Column 3 (π i0 ), is a transformation of the α i0 parameter which denotes the probability of endorsing each SMFQ item for an individual with a latent trait score of
zero, i.e. at the median/mean/modal value on the latent trait. The parameters have been estimated using full information marginal maximum likelihood estimation (modified EM algorithm) in TWOMISS
software.
Test Information Function (TIF)
We first report a graphic representation of the psychometric performance of the test as a whole (Fig. 1)—the
TIF. The TIF is derived from the inverse of the posterior
standard deviation of the latent trait estimates, when the
factor scores (or latent traits estimates) are estimated using
Bayesian (EAP) estimation. The TIF plots a function of
the standard error. Taking the reciprocal of variance or
standard deviation provides a humped plot with higher
values indicating regions of precise measurement (small
standard errors, relative to other regions)—this is “test information.” As expected, the curve dips sharply at the end
points where the SMFQ items provide little information.
Figure 1 clearly shows that the most information (and
therefore highest precision of measurement) is provided
by the SMFQ around 1.5 standard deviations above the
mean (0) on the latent trait.
Item Characteristic Curves (ICCs)
In Fig. 2, we represent the model results in terms of
ICCs or “tracelines” in which the impact of the two IRT
parameters are easily visible in relation to each other for
each individual item. ICCs are S-shaped functions, plotted
as a function of latent trait depression scores.
All items function at more or less the same level on
the latent trait. Importantly, all items are located towards
the more severe end—to the right of the figures. Both
Figs. 1 and 2 suggest that a child located between 1 and 2
standard deviations above (worse) the population mean on
the latent trait would have a 50% probability of endorsing
the SMFQ items. We can also see that for a child at the
mean (or median) latent trait value (0) the probability of
endorsing any item is very low. This is also reflected in
the entries in the last column of Table IV.
Although items 3 (tired) and 4 (restless) (top right
two ICCs in Fig. 2) also function at the severe end of the
latent trait, they show more shallow slopes. This indicates
lower discriminating power with respect to the latent trait
An IRT and CDFA Analysis of the SMFQ
387
Fig. 1. Test characteristic curve and test information function for the SMFQ in a community sample of 7- through 11-year-olds.
(depression). These are thus the least sensitive items for
measuring depression.
Items showing the highest commonality/lowest
prevalence parameters were items 9: bad person
(α i0 = − 4.3058) and 13: did everything wrong
(α i1 = − 4.2836). Items 5: felt no good; 8: hated self
and 13 showed the highest discrimination parameters (α i1 = 2.4916; α i1 = 2.4440 and α i1 = 2.4444,
respectively).
The Effect of Age
We examined the possibility of item bias (DIF) using an extension to IRT modelling referred to as MIMIC
modelling (see data analytic strategy). Results showed
that the ratio of estimate to standard error was <1.96 for
the direct effect of the covariate on all items, indicating
no significant item bias effects.
DISCUSSION
The current study is the first investigation in which
a combined IRT/CDFA analysis, including MIMIC modelling, of the SMFQ has been carried out in 7–11-year-old
children ascertained from the community. Methods for
summarizing item responses were originally developed in
the field of psychometrics and have been widely applied
for the purposes of educational testing. They have found
increasing application in medical (epidemiological) and
health care (survey) research for the assessment of psychopathology, but have rarely been applied to evaluate
clinical psychopathological assessments intended for use
in children.
The advantages of IRT for abnormal psychology
measurement are quite substantial (Cooke & Michie,
1997; Duncan-Jones et al., 1986; Embretson & Reise,
2000; Rouse et al., 1999; Santor et al., 1994). In scale
construction and evaluation, IRT item analysis provides
information that can be used for evaluating which items
from a large pool should be used together to comprise
a scale that is fit for that purpose (Waller, Tellegen,
McDonald, & Lykken, 1996). IRT results can be used
388
Sharp, Goodyer, and Croudace
Fig. 2. Item characteristic curves for the 13 items of the SMFQ. Items 1–13 (panelled from left to right,
first item top left). Notes: Axis labels removed for clarity of presentation. The x-axis is the estimated
latent trait score which is distributed as a standard normal distribution; The x-axis ranges from − 3
to + 3. The y-axis is the probability that the SMFQ symptom is endorsed; the y-axis ranges from a
minimum value of 0 to maximum value of 1.
Table IV. Confirmatory Factor Analyses using Mplus that Gives IRT Results for Categorical Response SMFQ Items (Binary Recoded)
CDFA
IRT
Item
Threshold (tau)
Loading (lambda)
Residual variance
(1 − lambda2 )
Z test for loading
(estimate/SE)
Discrimination
parameter a
Intercept
parameter b
1. Miserable
2. Not enjoy
3. Tired
4. Restless
5. Felt no good
6. Cried a lot
7. Poor
concentration
8. Hated self
9. Bad person
10. Felt lonely
11. Unloved
12. Never be as good
13. Everything wrong
1.24
1.31
0.93
0.59
1.20
1.34
0.92
0.68
0.54
0.53
0.40
0.79
0.60
0.72
0.53
0.70
0.71
0.83
0.36
0.63
0.48
12.96
8.19
9.21
6.79
19.74
9.73
15.16
1.05
0.76
0.68
0.48
1.42
0.89
1.13
1.74
2.22
1.66
1.35
1.50
2.06
1.26
1.20
1.55
0.97
1.35
1.03
1.42
0.77
0.70
0.67
0.72
0.74
0.77
0.39
0.50
0.53
0.47
0.44
0.40
18.97
12.43
14.14
15.07
17.45
15.99
1.39
1.22
1.00
1.26
1.18
1.38
1.51
2.04
1.41
1.76
1.37
1.79
Notes. Parameters (loadings/lambda and thresnolds/tau) estimated using robust weighted least squares estimation (weighted lease squares
estimation with mean– and variance-adjusted chi-square test statistic). Model fit under WLSMV: Chi-square value under robust WLS (mean
and variance adjusted) = 55.197, df = 46; p = 0.16; Comparative Fit Index = 0.992; Tucker Lewis Index = 0.994; Root mean square error
of approximation = 0.018; Weighted root mean square residual = 0.833. Tau: Mplus thresholds; Lambda: Mplus factor loadings; IRT aparameter: Discrimination parameter from response function parameterisation of CDFA, related to the factor loading (lambda); IRT Intercept
(b-parameter): x-axis value where p(item) = 0.5 in Fig. 2; CDFA denotes parameters from ‘underlying variable parameterisation of the two
parameter probit-probit IRT model; IRT denotes parameters under response function parameterisation of the two parameter probit-probit IRT
model.
An IRT and CDFA Analysis of the SMFQ
to estimate a person’s trait level (latent trait score), which
may be more accurate than summing unweighted individual item scores and can be calculated when there are
partially missing data. IRT results are useful for determining whether the items included in a candidate scale
exhibit differential item functioning for two populations
(Cooke & Michie, 1997; Rouse et al., 1999). This particular advantage may be especially relevant in studies
where developmental differences in the phenomenology
of depression are the focus of interest (see Weiss & Garber, 2003 for a review of such studies). Lastly, using IRT
technologies, there is potential for a clinical scale to be
substantially shortened through the adoption of an adaptive testing approach (Gardner et al., 2004). With adaptive
testing (computerised adaptive testing—CAT) a software
program uses IRT parameters to select items in a sequence
that optimally gain information on the severity score for
each respondent, by tailoring questions to the likely level
of the individual’s score. In summary, latent trait modelling techniques (like IRT and CDFA) provide a closer
approximation of the structural model underpinning psychiatric data than traditional linear methods (Cooke &
Michie, 1997), thereby giving a better representation of
empirical data and ultimately uncovering new aspects of
a familiar instrument (Duncan-Jones et al., 1986).
By applying an IRT/CDFA approach we demonstrated, in line with Angold et al.’s findings (2002), that
the SMFQ measures a unidimensional construct of depressive symptoms. Other studies have testified to the
external validity of the SMFQ (see Table I). Based on
these studies, and the face validity of SMFQ items, we
assumed the latent trait underlying SMFQ items to be
that of depression. However, in the absence of external
validity data we cannot conclude definitively that the
construct we studied was in fact depression. Notwithstanding this limitation, the current results support the
SMFQ as a highly homogenous/unidimensional measure.
Homogeneity/unidimensionality of a measure maximizes
the ability to discriminate between diagnostically distinct
groups (Costello & Angold, 1988) and thus speaks to the
validity of the measure.
High correlations between SMFQ items in this sample suggest good internal consistency for the scale in
younger children. Factor loadings for all SMFQ items
were high, indicating that the items were highly discriminating with respect to the latent trait. Item distribution
along the latent trait continuum was also consistent with
the goals of the instrument to be used for screening purposes. All items were found to function at the severe
end of the latent trait, with none being useful to measure individuals at the 50th percentile in the population
(mean, median—the midpoint of the latent trait). Accord-
389
ing to these results, the SMFQ may not be appropriate
for use in community studies where the interest lies in
variation in average (mental) health, because this would
require items to be more widely distributed across the
range of the latent trait. As such, the SMFQ may be more
appropriate for detecting children ages 7–11 who are
likely to report high levels of depressive symptoms at the
time of measurement. Despite the low response rate in the
current study, which may affect the generalizability of our
findings, the current study indicates with more appropriate statistical modelling techniques that the SMFQ does
what it was designed to do (Angold, Costello, Pickles, &
Winder, 1987).
An interesting next step would be to carry out IRT
analyses of the SMFQ in a clinical sample with a current
diagnosis of depression. This should yield a similar pattern of item loadings but the score distributions would be
skewed toward the depressed end of the latent trait. In our
sample, most children endorsed zero scores on the SMFQ,
which is what is expected for community samples.
Our analyses showed that two items function a little differently than the others (3: tired and 4: restless).
These items were characterised by ICCs that exhibited
shallower slopes and leftmost thresholds. They therefore
offer almost no discriminating power at the most severe
end of the depressive latent trait but would contribute to
the discrimination of individuals with lower scores. This
does not, however, provide a strong case for shortening
the SMFQ any further. Although these items exhibited
shallower slopes and lower thresholds in comparison with
other items, their psychometric properties are not sufficiently different from the other items to exclude them.
In contrast, items 8: hated self and 13: did everything
wrong showed most discriminating power, whilst items
9: bad person and 13: did everything wrong functioned
at the most severe end of the latent trait. This suggest
that these questions may be the most useful to ask if a
clinician or researcher is interested in identifying a child
with depression with expediency.
The information gained from these analyses regarding individual item performance is also of theoretical and
substantive interest. Items 3: tired and 4: restless which
did not load as highly (but still sufficiently to retain them)
on to the latent trait can be seen as less central to the trait
for children ages 7–11 compared with items 8: hated self,
9: bad person, 11: unloved, 12: never be as good and 13:
did everything wrong that load highly. Within this age
range, cognitive symptoms of depression such as items 8,
9, 11 and 12 show higher factor loadings, and therefore
more discriminatory power at the severe end of the latent
trait compared with “somatic” symptoms like items 3 and
4. These results are in line with previous research showing
390
that both cognitive and somatic symptoms in the long form
of the MFQ are essential to the construct of depression
in children and adolescents (Kent et al., 1997), but may
differ in their prevalence and/or loading on a single factor
of severity. The current results showed somatic symptoms
of depression (tired and restless) contributed less than cognitive symptoms to the underlying unidimensional latent
trait of depression in 7–11–year-old children.
In summary, the IRT analyses conducted here suggest
that the SMFQ: (1) is a unidimensional or homogeneous
measure; (2) has items showing relatively good discrimination at the severe end of the latent trait, indicating that
the instrument, in terms of its internal construct validity, is
an adequate screening measure for depressive symptoms
in a community sample of children ages 7–11-years-old;
(3) although this study lacked an adolescent comparison
sample, these data suggest that the SMFQ is not biased
with respect to age from 7- to 11-years-old and is suitable
for use with children as young as 7.
ACKNOWLEDGMENTS
The authors are grateful to all the families and
schools who participated. We also wish to thank Sarah
Moore, Heather Brown, Penny Hazell and Maria Loades
for helping with data collection and entry, and Dr. Melanie
Merricks and Dr. Carol Stott for valuable discussion. Carla
Sharp was supported by an NHS Post-Doctoral Fellowship, University of Cambridge, UK. Tim Croudace was
supported by a Dept of Health Career Scientist Award
(Public Health).
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